Running water making some people pee is because of power of suggestion.
The running water tricks your mind into your bladder needing to release the pee it already has in it.
Whenever I hear running water or if I wash my hands before I need to fully pee then I will need to pee within a few seconds.
When I was a kid my younger brother would also wet his pants when he heard running water.
Because he would be in the kitchen at the table and be eating and holding in his pee already when he was potty training.
Then he would not make it to the bathroom in time to pee.
You can use running water to get some toddlers to pee when they are sitting on the toilet and potty training.
It worked on my nephew when he was 3 years old and refusing to use the toilet and would hold it in until he got a diaper back on.
He would immediately pee in the diaper once his mom put him back in his diaper.
But after his mom started turning water on in the bathroom sink while he was sitting on the toilet he would release his bladder and pee into the toilet.